• Some irreducibility and indecomposability results for truncated binomial polynomials of small degree

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      https://www.ias.ac.in/article/fulltext/pmsc/127/01/0045-0057

    • Keywords

       

      Truncated binomial expansion; irreducibility; indecomposability.

    • Abstract

       

      In this paper, we show that the truncated binomial polynomials defined by $P_{n, k}(x)=\sum^k_{j=0}({n\choose j})x^j$ are irreducible for each $k\leq 6$ and every $n\geq k+2$. Under the same assumption $n\geq k+2$, we also show that the polynomial $P_{n, k}$ cannot be expressed as a composition $P_{n, k}(x)=g(h(x))$ with $g\in\mathbb{Q}[x]$ of degree at least 2 and a quadratic polynomial $h\in\mathbb{Q}[x]$. Finally, we show that for $k\geq 2$ and $m, n\geq k+1$ the roots of the polynomial $P_{m, k}$ cannot be obtained from the roots of $P_{n, k}$, where $m\neq n$, by a linear map.

    • Dates

       
  • Proceedings – Mathematical Sciences | News

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